functions representable as a sum of two permutations

Question

I am trying to prove that for every function $f:\mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ satisfying $\sum_if(i)=0$, there exist permutations $\pi_1, \pi_2:\mathbb{Z}/n\mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$ such that $f=\pi_1+\pi_2$. This is supposedly true, but I don't see why.

Answer 1

$\pi_2$ is a permutation iff $-\pi_2$ is a permutation. Thus, the problem is equivalent to proving that every such $f$ is the difference of two permutations. In A Combinatorial Problem on Abelian Groups (Proc. Amer. Math. Soc. 3 (1952), 584–587), M. Hall showed that in an abelian group of finite order $n$ (and hence in $\mathbb Z/n\mathbb Z$) any $n$-tuple of values that sums to $0$ is the difference of two permutations of the group elements.